Russia-US relations will see 'new chapter' but don't expect a quick end to sanctions, says CEO of Russia's largest bank

A man walks past a branch of Sberbank Rossii on March 7, 2014 in Simferopol, Ukraine.

America and Russia will likely see a thawing of relations in the coming months, according to the chief executive of Sberbank, amid high expectations that President-elect Donald Trump can re-establish close ties with his Russian counterpart.

Herman Gref, the CEO of Russia's largest bank and a former lawmaker in the country, said it was too early to comment the details of that relationship but predicted that it could be "new beginning" for the two countries.

"It's a new page, a new chapter between Russia and the U.S.," he said, hoping that it would be a return to the ties that former President George W. Bush and Vladimir Putin enjoyed in the last decade.

There had been low expectations in Russia for Bush before he was elected, after the close bonds that Bill Clinton had enjoyed with the country, Gref explained.

"We have a good chance to change our relations and I think that from the Russian side, maybe altogether we are ready to make huge steps in this direction," he told CNBC at the World Economic Forum in Davos.

Russia is still currently on the sharp end of tough international sanctions from the U.S. due to its annexation of Crimea and its alleged role in the destabilization of eastern Ukraine. Despite his optimism, Gref predicted that these penalties would not end this year.